Worked examples
The question
Question: find the parametric equations of the line through (1, 2, 3) and (4, 6, 8).
(1, 2, 3) and (4, 6, 8)
1 Find the direction
Draw the three axes.
Start at (1, 2, 3).
The line also goes through (4, 6, 8).
The arrow from P to Q shows the direction.
The direction is end minus start, part by part.
v = ⟨4 − 1, 6 − 2, 8 − 3⟩
4 minus 1 is 3.
6 minus 2 is 4.
8 minus 3 is 5.
So v = ⟨3, 4, 5⟩.
v = ⟨3, 4, 5⟩
2 Write the equations
Start at the point, then add t times the direction.
t is any number: each t gives one point of the line.
r(t) = ⟨1, 2, 3⟩ + t⟨3, 4, 5⟩
point + t × direction
Careful!
Only the direction is multiplied by t.
The start point stays as it is.
Careful: the point is added, the direction gets the t
Read each part: x = 1 + 3t.
x = 1 + 3t
Then: y = 2 + 4t.
y = 2 + 4t
And: z = 3 + 5t.
z = 3 + 5t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (1, 2, 3), the start.
Right!
t = 0: (1 + 3(0), 2 + 4(0), 3 + 5(0)) = (1, 2, 3)
t = 1 gives (4, 6, 8), the other point.
Right!
t = 1: (4, 6, 8) = (4, 6, 8)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
▶ Watch it solved
The question
Question: find the parametric equations of the line through (5, 1, 0) and (2, 3, 1).
(5, 1, 0) and (2, 3, 1)
1 Find the direction
Draw the three axes.
Start at (5, 1, 0).
The line also goes through (2, 3, 1).
The arrow from P to Q shows the direction.
The direction is end minus start, part by part.
v = ⟨2 − 5, 3 − 1, 1 − 0⟩
2 minus 5 is −3.
3 minus 1 is 2.
1 minus 0 is 1.
So v = ⟨−3, 2, 1⟩.
v = ⟨−3, 2, 1⟩
2 Write the equations
Start at the point, then add t times the direction.
t is any number: each t gives one point of the line.
r(t) = ⟨5, 1, 0⟩ + t⟨−3, 2, 1⟩
point + t × direction
Careful!
Only the direction is multiplied by t.
The start point stays as it is.
Careful: the point is added, the direction gets the t
Read each part: x = 5 − 3t.
x = 5 − 3t
Then: y = 1 + 2t.
y = 1 + 2t
And: z = t.
z = t
3 Check
Here is the whole line.
Let's check.
t = 0 gives (5, 1, 0), the start.
Right!
t = 0: (5 − 3(0), 1 + 2(0), (0)) = (5, 1, 0)
t = 1 gives (2, 3, 1), the other point.
Right!
t = 1: (2, 3, 1) = (2, 3, 1)
Let's look back.
First: find the direction.
Then: write the equations.
Last: check.
▶ Watch it solved
The question
Question: find the symmetric equations of the line through (1, −2, 3) and (3, 2, 4).
(1, −2, 3) and (3, 2, 4)
1 Find the direction
Draw the three axes.
Start at (1, −2, 3).
The line also goes through (3, 2, 4).
The arrow from P to Q shows the direction.
The direction is end minus start, part by part.
v = ⟨3 − 1, 2 − (−2), 4 − 3⟩
3 minus 1 is 2.
2 minus (−2) is 4.
4 minus 3 is 1.
So v = ⟨2, 4, 1⟩.
v = ⟨2, 4, 1⟩
2 Write the equations
Start at the point, then add t times the direction.
t is any number: each t gives one point of the line.
r(t) = ⟨1, −2, 3⟩ + t⟨2, 4, 1⟩
point + t × direction
Careful!
Only the direction is multiplied by t.
The start point stays as it is.
Careful: the point is added, the direction gets the t
Read each part: x = 1 + 2t.
x = 1 + 2t
Then: y = −2 + 4t.
y = −2 + 4t
And: z = 3 + t.
z = 3 + t
3 Solve each for t
Solve each equation for t: take away the start number, then divide by the direction number.
t = x − 12, t = y + 24, t = z − 3
All are the same t, so they are equal to each other: the symmetric equations.
= x − 12 = y + 24 = z − 3
4 Check
Here is the whole line.
Let's check.
t = 0 gives (1, −2, 3), the start.
Right!
t = 0: (1 + 2(0), −2 + 4(0), 3 + (0)) = (1, −2, 3)
t = 1 gives (3, 2, 4), the other point.
Right!
t = 1: (3, 2, 4) = (3, 2, 4)
Let's look back.
First: find the direction.
Then: write the equations.
Then: solve each for t.
Last: check.
Answer = x − 12 = y + 24 = z − 3
▶ Watch it solved
